The binary linear code \(H^\bot_{m,2}\), \(m > 2\), of length \(\binom{m}{2}\) represented by the generator matrix \(H_{m,2}\) consisting of all distinct column strings of length \(m\) and Hamming weight \(2\) is considered. A parity-check matrix \(H^\bot_{m,2}\) is assigned to the code \(H^\bot_{m,2}\). The code \(H_{m,2,3}\), \(m > 3\), of length \(\binom{m}{2} + \binom{m}{3}\) represented by the parity-check matrix \(H_{m,2,3}\) consisting of all distinct column strings of length \(m\) and Hamming weight two or three is also considered. It is shown that \(H^\bot_{m,2}\) and \(H_{m,2,3}\) are optimal stopping redundancy codes, that is for each of these codes the stopping distance of the associated parity-check matrix is equal to the minimum Hamming distance of the code, and the rows of the parity-check matrix are linearly independent. Explicit formulas determining the number of stopping sets of arbitrary size for these codes are given.

For a finite group \(G\) and subsets \(T_1, T_2\) of \(G\), the Bi-Cayley digraph \(D = (V(D), E(D)) = D(G, T_1, T_2)\) of \(G\) with respect to \(T_1\) and \(T_2\) is defined as the bipartite digraph with vertex set \(V(D) = G \times \{0, 1\}\), and for \(g_1, g_2 \in G\), \(((g_1, 0), (g_2, 1)) \in E(D)\) if and only if \(g_2 = t_1 g_1\) for some \(t_1 \in T_1\), and \(((g_1, 1), (g_2, 0)) \in E(D)\) if and only if \(g_1 = t_2 g_2\) for some \(t_2 \in T_2\). If \(|T_1| = |T_2| = k\), then \(D\) is \(k\)-regular. In this paper, the spectra of Bi-Circulant digraphs are determined. In addition, some asymptotic enumeration theorems for the number of directed spanning trees in Bi-Circulant digraphs are presented.

The genus of a graph \(G\), denoted by \(\gamma(G)\), is the minimum genus of an orientable surface in which the graph can be embedded. In the paper, we use the Joint Tree Model to immerse a graph on the plane and obtain an associated polygon of the graph. Along the way, we construct a genus embedding of the edge disjoint union of \(K\) and \(H\), and solve Michael Stiebitz’s proposed conjecture: Let \(G\) be the edge disjoint union of a complete graph \(K\) and an arbitrary graph \(H\). Let \(H’\) be the graph obtained from \(H\) by contracting the set \(V(X)\) to a single vertex. Then

\[\gamma(K) + \gamma(H’) \leq \gamma(G).\]

We investigate brother avoiding round robin doubles tournaments and construct several infinite families. We show that there is a BARRDT(\(x\)) that is not a SAMDRR(\(n\)) for all \(n > 4\).

A digraph \(D(V, E)\) is said to be graceful if there exists an injection \(f: V(G) \to \{0, 1, \ldots, |E|\}\) such that the induced function \(f’: E(G) \to \{1, 2, \ldots, |E|\}\) which is defined by \(f'(u, v) = [f(v) – f(u)] \pmod{|E| + 1}\) for every directed edge \((u, v)\) is a bijection. Here, \(f\) is called a graceful labeling (graceful numbering) of \(D(V, E)\), while \(f’\) is called the induced edge’s graceful labeling of \(D\). In this paper, we discuss the gracefulness of the digraph \(n – \overrightarrow{C}_m\), and prove that \(n – \overrightarrow{C}_m\) is a graceful digraph for \(m = 4, 6, 8, 10\) and even \(n\).

In this note, we consider relative difference sets with the parameter \((m, 2, m-1, \frac{m-2}{2})\) in a group \(G\) relative to a subgroup \(N\). In the splitting case, \(G = H \times N\), we give a lower bound for the size of the commutator group \(H’\), and we show that \(H\) cannot have a homomorphic image which is generalized dihedral. In the non-splitting case, we prove that there is no \((2n, 2, 2n-1, n-1)\) relative difference set in a generalized dihedral group of order \(4n\), \(n > 1\).

Let \(P_n\) be a path with \(n\) vertices. \(P_n^k\), the \(k\)-th power of the path \(P_n\), is a graph on the same vertex set as \(P_n\), and the edges that join all vertices \(x\) and \(y\) if and only if the distance between them is at most \(k\). In this paper, the crossing numbers of \(P_n^k\) are studied. Drawings of \(P_n^k\) are presented and proved to be optimal for the case \(n \leq 8\) and for the case \(k \leq 4\).

A graph is said to be locally grid if the structure around each of its vertices is a \(3 x 3\) grid. As a follow up of the research initiated in \([8]\) and \([9]\) we prove that most locally grid graphs are uniquely determined by their Tutte polynomial.

Let \(P(G, \lambda)\) be the chromatic polynomial of a graph \(G\). A graph \(G\) is chromatically unique if for any graph \(H\), \(P(H, \lambda) = P(G, \lambda)\) implies \(H\) is isomorphic to \(G\). In his Ph.D. thesis, Zhao [Theorems 5.4.2 and 5.4.3] proved that for any positive integer \(t \geq 3\), the complete \(t\)-partite graphs \(K(p – k, p, p, \ldots, p)\) with \(p \geq k+2 \geq 4\) and \(K(p-k, p – 1, p, \ldots, p)\) with \(p \geq 2k \geq 4\) are chromatically unique. In this paper, by expanding the technique employed by Zhao, we prove that the complete \(t\)-partite graph \(K(p-k,\underbrace{ p -1, \ldots, p-1}, \underbrace{p, \ldots, p})\) is chromatically unique for integers \(p \geq k+2 \geq 4\) and \(t \geq d+3 \geq 3\).

We present a block diagonalization method for the adjacency matrices of two types of covering graphs. A graph \(Y\) is a covering graph of a base graph \(X\) if there exists an onto graph map \(\pi: Y \to X\) such that for each \(x \in X\) and for each \(y \in \{y \mid \pi(y) = x\}\), the collection of vertices adjacent to \(y\) maps onto the collection of vertices adjacent to \(x \in X\). The block diagonalization method requires the irreducible representations of the Galois group of \(Y\) over \(X\). The first type of covering graph is the Cayley graph over the finite ring \(\mathbb{Z}/p^n\mathbb{Z}\). The second type of covering graph resembles large lattices with vertices \(\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}\) for large \(n\). For one lattice, the block diagonalization method allows us to obtain explicit formulas for the eigenvalues of its adjacency matrix. We use these formulas to analyze the distribution of its eigenvalues. For another lattice, the block diagonalization method allows us to find non-trivial bounds on its eigenvalues.

Broadcast domination in graphs is a variation of domination in which different integer weights are allowed on vertices and a vertex with weight \(k\) dominates its distance \(k\)-neighborhood. A distribution of weights on vertices of a graph \(G\) is called a dominating broadcast, if every vertex is dominated by some vertex with positive weight. The broadcast domination number \(\gamma_b(G)\) of a graph \(G\) is the minimum weight (the sum of weights over all vertices) of a dominating broadcast of \(G\). In this paper, we prove that for a connected graph \(G\), \(\gamma_b(G) \geq \lceil{2\text{rad}(G)}/{3}\rceil\). This general bound and a newly introduced concept of condensed dominating broadcast are used in obtaining sharp upper bounds for broadcast domination numbers of three standard graph products in terms of broadcast domination numbers of factors. A lower bound for a broadcast domination number of the Cartesian product of graphs is also determined, and graphs that attain it are characterized. Finally, as an application of these results, we determine exact broadcast domination numbers of Hamming graphs and Cartesian products of cycles.

The semigirth \(\gamma\) of a digraph \(D\) is a parameter related to the number of shortest paths in \(D\). In particular, if \(G\) is a graph, the semigirth of the associated symmetric digraph \(G^*\) is \(\ell(G^*) = \lfloor {g(G) – 1}/{2} \rfloor\), where \(g(G)\) is the girth of the graph \(G\). In this paper, some bounds for the minimum number of vertices of a \(k\)-regular digraph \(D\) having girth \(g\) and semigirth \(\ell\), denoted by \(n(k, g; \ell)\), are obtained. Moreover, we construct a family of digraphs which achieve the lower bound for some particular values of the parameters.

For a graph \(G\), let \(\mathcal{D}(G)\) be the set of all strong orientations of \(G\). Define the orientation number of \(G\), \(\overrightarrow{d}(G) = \min\{d(D) \mid D \in \mathcal{D}(G)\}\), where \(d(D)\) denotes the diameter of the digraph \(D\). In this paper, it has been shown that \(\overrightarrow{d}(G \times H) = d(G)\), where \(\times\) denotes the tensor product of graphs, \(H\) is a special type of circulant graph, and the diameter, \(d(G)\), of \(G\) is at least \(4\). Some interesting results have been obtained using this result. Further, it is shown that \(d(P_r \times K_s) = d(P_r)\) for suitable \(r\) and \(s\). Moreover, it is proved that \(\overrightarrow{d}(C_r \times K_s) = d(C_r)\) for appropriate \(r\) and \(s\).

We consider some partitions where even parts appear twice and some where evens do not repeat. Further, we offer a new partition theoretic interpretation of two mock theta functions of order \(8\).

A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. The purpose of this paper is to generalize some known theorems and results of cordial graphs. Specifically, we show that certain combinations of paths, cycles, and stars are cordial.

An edge-magic total labeling on a graph with \(p\) vertices and \(q\) edges is defined as a one-to-one map taking the vertices and edges onto the integers \(1, 2, \ldots, p+q\) with the property that the sum of the labels on an edge and of its endpoints is constant, independent of the choice of edge. The magic strength of a graph \(G\), denoted by \(emt(G)\), is defined as the minimum of all constants over all edge-magic total labelings of \(G\). The maximum magic strength of a graph \(G\), denoted by \(eMt(G)\), is defined as the maximum constant over all edge-magic total labelings of \(G\). A graph \(G\) is called weak magic if \(eMt(G) – emt(G) > p\). In this paper, we study some classes of weak magic graphs.

In the first part of this paper, we present a generalization of complete graph factorizations obtained by labeling the graph vertices by natural numbers. In this generalization, the vertices are labeled by elements of an arbitrary group \(G\), in order to achieve a \(G\)-transitive factorization of the graph.

Vertex colorings of Steiner systems \(S(t,t+1,v)\) are considered in which each block contains at least two vertices of the same color. Necessary conditions for the existence of such colorings with given parameters are determined, and an upper bound of the order \(O(\ln v)\) is found for the maximum number of colors. This bound remains valid for nearly complete partial Steiner systems, too. In striking contrast, systems \(S(t,k,v)\) with \(k \geq t+2\) always admit colorings with at least \(c\cdot v^\alpha\) colors, for some positive constants \(c\) and \(\alpha\), as \(v\to\infty\).

Cwatsets were originally defined as subsets of \(\mathbb{Z}_2^d\) that are “closed with a twist.” Attempts have been made to generalize them, but the generalizations have failed to produce notions of subcwatset and quotient cwatset that behave naturally.

We present a new, abstract definition that appears to avoid these problems. The relationship between this new definition and its predecessor is similar to that between the abstract definition of “group” and its original meaning as a set of permutations. To justify the broader definition, we use small cancellation theory to prove a result analogous to the statement that every group is isomorphic to some permutation group. After developing the notion of a quotient cwatset, we prove an analogue of the First Homomorphism Theorem.

In this paper, we consider a class of recursively defined, full binary trees called Lucas trees and investigate their basic properties. In particular, the distribution of leaves in the trees will be carefully studied. We then go on to show that these trees are \(2\)-splittable, i.e., they can be partitioned into two isomorphic subgraphs. Finally, we investigate the total path length and external path length in these trees, the Fibonacci trees, and other full \(m\)-ary trees.

A tree \(T\) with \(n\) vertices and a perfect matching \(M\) is strongly graceful if \(T\) admits a graceful labeling \(f\) such that \(f(u)+f(v) = n-1\) for every edge \(uv \in M\). Broersma and Hoede \([5]\) conjectured that every tree containing a perfect matching is strongly graceful in \(1999\). We prove that a tree \(T\) with diameter \(D(T) \leq 5\) supports the strongly graceful conjecture on trees. We show several classes of basic seeds and some constructive methods for constructing large scales of strongly graceful trees.

In a previous paper, the first author introduced two classes of generalized Stirling numbers, \(s_m(n,k,p), S_m(n,k,p)\) with \(m = 1\) or \(2\), called \(p\)-Stirling numbers. In this paper, we discuss their determinant properties.

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the problem of PI index with respect to some simple pericondensed hexagonal systems and we solve it completely.

As a part of the author’s work of enumerating the edge-forwarding indices of Frobenius graphs, I give a class of valency four Frobenius graphs derived from the Frobenius groups \(\mathbb{Z}_{4n^2+1} \rtimes \mathbb{Z}_4\). Following the method of Fang, Li and Praeger, some properties including the diameter and the type of this class of graphs are given (Theorem \(3.2\)).

We address the problem of determining all sets which form minimal covers of maximal cliques for interval graphs. We produce an algorithm enumerating all minimal covers using the C-minimal elements of the interval order, as well as an independence Metropolis sampler. We characterize maximal removable sets, which are the complements of minimal covers, and produce a distinct algorithm to enumerate them. We use this last characterization to provide bounds on the maximum number of minimal covers for an interval order with a given number of maximal cliques, and present some simulation results on the number of minimal covers in different settings.

A directed triple system of order \(v\), denoted by DTS\((v)\), is a pair \((X,\mathcal{B})\) where \(X\) is a \(v\)-set and \(\mathcal{B}\) is a collection of transitive triples on \(X\) such that every ordered pair of \(X\) belongs to exactly one triple of \(\mathcal{B}\). A DTS\((v)\) is called pure and denoted by PDTS\((v)\) if \((x,y,z) \in \mathcal{B}\) implies \((z,y,x) \notin \mathcal{B}\). A large set of disjoint PDTS\((v)\) is denoted by LPDTS\((v)\). In this paper, we establish the existence of LPDTS\((v)\) for \(v \equiv 0,4 \pmod{6}\), \(v\geq 4\).

We extend and give short proofs of some recent results regarding some classes of rational difference equations.

The skewness \(sk(G)\) of a graph \(G = (V, E)\) is the smallest integer \(sk(G) \geq 0\) such that a planar graph can be obtained from \(G\) by the removal of \(sk(G)\) edges. The splitting number \(sp(G)\) of \(G\) is the smallest integer \(sp(G) \geq 0\) such that a planar graph can be obtained from \(G\) by \(sp(G)\) vertex splitting operations. The vertex deletion \(vd(G)\) of \(G\) is the smallest integer \(vd(G) \geq 0\) such that a planar graph can be obtained from \(G\) by the removal of \(vd(G)\) vertices. Regular toroidal meshes are popular topologies for the connection networks of SIMD parallel machines. The best known of these meshes is the rectangular toroidal mesh \(C_m \times C_n\), for which is known the skewness, the splitting number and the vertex deletion. In this work we consider two related families: a triangulation \(T_{m,n}\) of \(C_m \times C_n\) in the torus, and an hexagonal mesh \(H_{m,n}\), the dual of \(\mathcal{T}_{C_m\times C_n}\) in the torus. It is established that \(sp(T_{m,n}) = vd(T_{m,n}) = sk(H_{C_m\times C_n}) = sp(\mathcal{H}_{C_m\times C_n}) = vd(\mathcal{H}_{m,n}) = \min\{m,n\}\) and that \(sk(\mathcal{T}_{C_m\times C_n}) = 2\min\{m, n\}\).

Exploiting the empirical observation that the probability of \(k\) fixed points in a Welch-Costas permutation is approximately the same as in a random permutation of the same order, we propose a stochastic model for the most probable maximal number of fixed points in a Welch-Costas permutation.

Let \(\gamma_c(G)\)be the connected domination number of \(G\) and \(\gamma_t(G)\) be the tree domination number of \(G\). In this paper, we study the connected domination number and tree domination of \(P(n,k)\), and show that \(\gamma_{tr}(P(n, 4)) = \gamma_c(P(n, 4)) = n-1\) for \(n \geq 17\), \(\gamma_{tr}(P(n, 6)) = \gamma_c(P(n, 6)) = n-1\) for \(n \geq 25\), and \(\gamma_{tr}(P(n,8)) = \gamma_c(P(n,8)) = n-1\) for \(n \geq 33\).

A cut \((A, B)\) (where \(B = V – A\)) in a graph \(G = (V, E)\) is called internal if and only if there exists a vertex \(x \in A\) that is not adjacent to any vertex in \(B\) and there exists a vertex \(y \in B\) such that it is not adjacent to any vertex in \(A\). In this paper, we present a theorem regarding the arrangement of cliques in a chordal graph with respect to its internal cuts. Our main result is that given any internal cut \((A, B)\) in a chordal graph \(G\), there exists a clique with \(\kappa(G) + 1\) vertices (where \(\kappa(G)\) is the vertex connectivity of \(G\)) such that it is (approximately) bisected by the cut \((A, B)\). In fact, we give a stronger result: For any internal cut \((A, B)\) of a chordal graph, and for each \(i\), \(0 \leq i \leq \kappa(G) + 1\), there exists a clique \(K_i\) such that \(|A \cap K_i| = \kappa(G) + 1\), \(|A \cap K_i| = i\), and \(|B \cap K_i| = \kappa(G) + 1- i\).

An immediate corollary of the above result is that the number of edges in any internal cut (of a chordal graph) should be \(\Omega(k^2)\) where \(\kappa(G)\). Prompted by this observation, we investigate the size of internal cuts in terms of the vertex connectivity of the chordal graphs. As a corollary, we show that in chordal graphs, if the edge connectivity is strictly less than the minimum degree, then the size of the mincut is at least \(\frac{\kappa(G)(\kappa(G) + 1)}{2}\), where \(\kappa(G)\) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to \(\kappa(G)\). This result is tight.

We determine the automorphism group and the spectrum of the folded hypercube. In addition, we define the Bi-folded hypercube and determine its spectrum.